FORCE 
thefe oe iad is the motion of a header 7 an Migs 
mediu t ge I 5) 
ie the 
axis of y be vertical, as there is ne other force but that of 
, 2 
—— = 6; 
gravity: X=o; Z=o0; Y = —g; therefore TP 
a ; and by integration 
GRaO Tp AnH 
x Ie ", dy _ 
We = 63 Fr c a =o gi. 
The firlt of thefe equations, divided by the fecond, gives by 
integration 
elt 
a to a ftraight line, eres “thews that the Pama 
or t rajeé xy i 
ftraight line, and that the curve is in a ee ae 
pafling through the axis of y 
d y 
= einige : : 
——=¢,—— =! — gt, by integration 
di OU ee a 
givex = ct; y= ct — ost 
No nea eae a need be added fince when ¢ = 
y= o determine the other two conftant quae 
tities, c and c!, at the commencement of metion, fince ——, 
i? t 
<4, are the velocities at every inftant, if the velocity V is 
d 
vertical, when ¢ = a, = = 0, and a = 
el But when the force producing the velocity V is 
in the direction of the angle 4, with the axis ot x, then when 
'¢= 0, ¢= V, cos. 6 an ee 
f the expreffions for x and y, one 
uniform motion, the other of uniformly va ried m 
The equations 
x = 
V, hence c= 0, 
is the eat of 
otion. ¢ 
gx 
2 
eliminating ¢ from thefe equations, we find y = — x — 
fubftituting for ¢ and <¢' their trigonometrical values, ie 
equation to the trajectory 
yu, tang. er ae 
V fubftitute its value 2¢4, (4 being the height due to — 
For 
the velocity V,) an 
y = x. tang. te 
4h. cos.’ ¢: 
equation to a parabola. 
red, itis to be foun 
In the folution of thefe pr aed we are to obferve, that 
dby # 
when the nature of the 
the curve, which reprefents the element of the {pace, and 
—— is known in terms of x or y 2 the equation to the 
cu 
T, in cs above a the proeatle be fuppofed to 
he of whofe refiftance is known, 
vie ‘svehigation of the curve becomes panache mo e difficult. 
The expreffions for - forces X, Y, Z, will no loge be 
of the fimple form will be 
more difficult to eliminate, and the fecieuueat Geet 
infinitely more complex : but ce i soblee principle will rez 
main pree cifely the fame. See JECTILE. 
‘The fame ‘opnciples apply to » the motion of bodie 
ove the particular cafes of this eopleics vill be 
given under Trajectory, the general nature of the m 
thod sith as follows. 
€ pro- 
eéte 
given point b s fom 
ance, and ict ee is ikewife known -at fome given 
diftance : from thefe data we are to tet i the principle 
by which the curve may be determi 
Let three ees soe - drawn through the’ 
centre of force ; gs let P reprefent the abfolute intenfity 
of the force at a v rawn 
oint M, in w t 
body is at that inftant, makes ~ each axis a certain 
,2 
angle; the cofines of thefe angles are—, 2, “3 the forces, 
therefore, that refult from the refolution of ‘the a P,. 
x 
and which are equivalent to it, are —, oe 
If, therefore, the element di# be fuppoed ee ‘hen, 
by the ee ormulz, (fince the forces tend to diminifh 
the co-ordinates, ) 
a’ x ay Py dy 
eas Ga Ga- Fo 
To integrate thefe equations, multiply the firft by y, the 
i by »,.and fubtraét the firft produét from the fecend :. 
x dy —y dx _ 
dt 
and. xdy —ydx=edt 
zdx—xdz=cldt 
paz —xzdy=cdt. 
Multiply thefe equations by Xs Yy % refpectively, and’ 
add, then cx + cly + ich nee belongs to 
the ahs aaa and indicates that this curve is a plane 
pafling-through the centre of fe Therefore, in the 
robl 
oblem, yiage 
ofx andy, Let F DM (fg. 1 
in which the moving body is fond at the end or the time #3 
A the centre of the — AP=~x, P » A 
The angle MA «=z, theri jclie-angled angle | for the ane 
Fascia of its one into polar co-ordinates, (fee Zua- 
lytic GEOMETRY, ), gives 
r= xt y’; aaa uy yr fin. wy 
ee aa, u-+-cos.udr 
dy =r cos. me udr 
See alfo Analytic Funcrion, where thefe differential equa- 
tions are explained. 
Therefore, x dy — —ydx=ardu; 
and r?du=c di. 
1 Ar peters orf. (xdy —ydx ) is the double of the 
ontained ih ae two radii vectores H;,. 
se sparuay be one of which is fixed : therefore, this area 
=3ct+ A, taki ing | the: radius A H as fixed and belong- 
ing f° “to M, when f = 
appears on whatever be the central a 
the a area eg ee will always be proportional to the tim 
3 
